This corresponds to Cameron and Trivedi’s equation (3.13) and thus
corresponds to the NB2 model in their terminology. The dispersion for this
model is (1 + alpha*mu_i), which depends on mu_i, hence the moniker
“mean dispersion”.

By comparison, nbreg, dispersion(constant) has the distribution of
g_i as

g_i ~ Gamma(mu_i/delta, delta)

where delta is the ancillary parameter. I could have easily called this
alpha and not delta, but nbreg uses delta to make the distinction
between both models clearer.

which (except for calling it delta instead of alpha) corresponds to Cameron
and Trivedi’s equation (3.11), and hence the NB1 model. For this
model, the dispersion is (1 + delta) and thus is constant over all
observations.

For both models, the dispersion is greater than one. This is why
nbreg serves its purpose of modeling data that exhibit dispersion
beyond that which can be handled using Poisson regression, which has
dispersion set to 1.